Well, Kaori actually calls him those things due to things that happened after meeting him.
He tended to eat burgers every time he was in a restaurant with her, and she even cut him off from burgers at one point. I also don't think he ever actually gave her a name. Not even Sensei. So, Hamburger man became his name. (Ironically enough, Hamburgers have been one thing he's claimed to love in his thoughts)
As for Human trafficker, it's because she saw him with Ami in one event, Rin in one event, and Sana in one event, all at separate times and hadn't seen the girls since, so she just assumed he was trafficking them.
As for the girls being related to him, I do think it's possible, but in a more all their mothers are angelic beings who procreated with Sensei to create pieces for this world kind of way. Hence why this happened with Sara:
Albeit, I doubt the mothers actually know what's going on, unless they are just good actors.